Abstract
A global optimization algorithm is presented for maximizing the sum of difference of convex functions ratios problem over nonconvex feasible region. This algorithm is based on branch and bound framework. To obtain a difference of convex programming, the considered problem is first reformulated by introducing new variables as few as possible. By using subgradient and convex envelope, the fundamental problem of estimating lower bound in the branch and bound algorithm is transformed into a relaxed linear programming problem which can be solved efficiently. Furthermore, the size of the relaxed linear programming problem does not change during the algorithm search. Lastly, the convergence of the algorithm is analyzed and the numerical results are reported.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Almogy, Y., Levin, O.: Parametric analysis of a multistage stochastic shipping problem. In: Proceedings of the 5th IFORS Conference, pp. 359–370 (1964)
Benson, H.P.: Using concave envelopes to globally solve the nonlinear sum of ratios problem. J. Glob. Optim. 22, 343–364 (2002)
Benson, H.P.: A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem. Eur. J. Oper. Res. 182, 597–611 (2007)
Colantoni, C.S., Manes, R.P., Whinston, A.: Programming, profit rates, and pricing decisions. Access Rev. 44, 467–481 (1969)
Craven, B.D.: Fractional Programming. Sigma Series in Applied Mathematics, vol. 4. Heldermann, Berlin (1988)
Dai, Y., Shi, J., Wang, S.: Conical partition algorithm for maximizing the sum of dc ratios. J. Glob. Optim. 31, 253–270 (2005)
Gotoh, J.Y., Konno, H.: Maximization of the ratio of two convex quadratic functions over a polytope. Comput. Optim. Appl. 20, 43–60 (2001)
Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Kluwer Academic, Boston (2000)
Konno, H., Inori, M.: Bond portfolio optimization by bilinear fractional programming. J. Oper. Res. Soc. Jpn. 32, 143–158 (1989)
Konno, H., Watanabe, H.: Bond portfolio optimization problems and their application to index tracking: a partial optimization approach. J. Oper. Res. Soc. Jpn. 39, 295–306 (1996)
Rao, M.R.: Cluster analysis and mathematical programming. J. Am. Stat. Assoc. 66, 622–626 (1971)
Schaible, S.: A note on the sum of a linear and linear fractional function. Nav. Res. Logist. Q. 24, 691–693 (1977)
Schaible, S.: Fractional programming. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization. Kluwer Academic, Dordrecht (1995)
Schaible, S.: Fractional programming with sums of ratios. In: Castagnoli, E., Giorgi, G. (eds.) Scalar and Vector Optimization in Economic and Financial Problems. Proceedings of the Workshop in Milan (1995)
Shen, P.P.: A simplicial branch and bound duality-bounds algorithm for the sum of convex-convex ratios problem. J. Comput. Appl. Math. 223, 145–158 (2009)
Stancu-Minasian, I.M.: Fractional Programming: Theory, Methods and Applications. Kluwer Academic, Dordrecht (1997)
Acknowledgements
We are very grateful to the anonymous reviewers for their valuable and insightful comments, which have aided us in improving the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors gratefully acknowledge the partial supports of the National Science Foundation Grant (10871130, 11171094) of China, the Ph.D. Foundation Grant (2009312711005) of Chinese Education Ministry.
Rights and permissions
About this article
Cite this article
Pei, Y., Zhu, D. Global optimization method for maximizing the sum of difference of convex functions ratios over nonconvex region. J. Appl. Math. Comput. 41, 153–169 (2013). https://doi.org/10.1007/s12190-012-0602-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-012-0602-8